Integrand size = 10, antiderivative size = 22 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \text {arcsinh}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)} \] Output:
-1/2*arcsinh(cot(x))-1/2*cot(x)*(csc(x)^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \sqrt {\csc ^2(x)} \left (-\csc ^2\left (\frac {x}{2}\right )-4 \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \log \left (\sin \left (\frac {x}{2}\right )\right )+\sec ^2\left (\frac {x}{2}\right )\right ) \sin (x) \] Input:
Integrate[(1 + Cot[x]^2)^(3/2),x]
Output:
(Sqrt[Csc[x]^2]*(-Csc[x/2]^2 - 4*Log[Cos[x/2]] + 4*Log[Sin[x/2]] + Sec[x/2 ]^2)*Sin[x])/8
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4140, 3042, 4610, 211, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (\cot ^2(x)+1\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\tan \left (x+\frac {\pi }{2}\right )^2+1\right )^{3/2}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \csc ^2(x)^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\sec \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle -\int \sqrt {\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{\sqrt {\cot ^2(x)+1}}d\cot (x)-\frac {1}{2} \sqrt {\cot ^2(x)+1} \cot (x)\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {1}{2} \text {arcsinh}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\cot ^2(x)+1}\) |
Input:
Int[(1 + Cot[x]^2)^(3/2),x]
Output:
-1/2*ArcSinh[Cot[x]] - (Cot[x]*Sqrt[1 + Cot[x]^2])/2
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {\cot \left (x \right ) \sqrt {\cot \left (x \right )^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (\cot \left (x \right )\right )}{2}\) | \(19\) |
default | \(-\frac {\cot \left (x \right ) \sqrt {\cot \left (x \right )^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (\cot \left (x \right )\right )}{2}\) | \(19\) |
risch | \(-\frac {i \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}+\sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )-\sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )\) | \(98\) |
Input:
int((cot(x)^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*cot(x)*(cot(x)^2+1)^(1/2)-1/2*arcsinh(cot(x))
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (16) = 32\).
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.14 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {2 \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} + \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right ) - \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right )}{4 \, \sin \left (2 \, x\right )} \] Input:
integrate((1+cot(x)^2)^(3/2),x, algorithm="fricas")
Output:
-1/4*(2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*(cos(2*x) + 1) + log(1/2*sqrt(2)*s qrt(-1/(cos(2*x) - 1))*sin(2*x) + 1)*sin(2*x) - log(-1/2*sqrt(2)*sqrt(-1/( cos(2*x) - 1))*sin(2*x) + 1)*sin(2*x))/sin(2*x)
\[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\int \left (\cot ^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}\, dx \] Input:
integrate((1+cot(x)**2)**(3/2),x)
Output:
Integral((cot(x)**2 + 1)**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (16) = 32\).
Time = 0.15 (sec) , antiderivative size = 300, normalized size of antiderivative = 13.64 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {4 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \left (x\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \] Input:
integrate((1+cot(x)^2)^(3/2),x, algorithm="maxima")
Output:
-1/4*(4*(cos(3*x) + cos(x))*cos(4*x) - 4*(2*cos(2*x) - 1)*cos(3*x) - 8*cos (2*x)*cos(x) + (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*log(cos( x)^2 + sin(x)^2 + 2*cos(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^ 2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos (2*x) - 1)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 4*(sin(3*x) + sin(x)) *sin(4*x) - 8*sin(3*x)*sin(2*x) - 8*sin(2*x)*sin(x) + 4*cos(x))/(2*(2*cos( 2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*s in(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (\frac {2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate((1+cot(x)^2)^(3/2),x, algorithm="giac")
Output:
1/4*(2*cos(x)/(cos(x)^2 - 1) - log(cos(x) + 1) + log(-cos(x) + 1))*sgn(sin (x))
Time = 8.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {\mathrm {asinh}\left (\mathrm {cot}\left (x\right )\right )}{2}-\frac {\mathrm {cot}\left (x\right )\,\sqrt {{\mathrm {cot}\left (x\right )}^2+1}}{2} \] Input:
int((cot(x)^2 + 1)^(3/2),x)
Output:
- asinh(cot(x))/2 - (cot(x)*(cot(x)^2 + 1)^(1/2))/2
\[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\int \sqrt {\cot \left (x \right )^{2}+1}d x +\int \sqrt {\cot \left (x \right )^{2}+1}\, \cot \left (x \right )^{2}d x \] Input:
int((1+cot(x)^2)^(3/2),x)
Output:
int(sqrt(cot(x)**2 + 1),x) + int(sqrt(cot(x)**2 + 1)*cot(x)**2,x)